Answer with Step-by-step explanation:
Let A and B be non- empty bounded subset of R
a.We have to find why
exist
If A and B are bounded set
Then there exist constant such that
and

Then , sup of A =b and sup of B=d
When a set is bounded then all elements lie in the set are lie between the constants s and t.
All elements are less than or equal to t then t is supremum of set.
Because both set are bounded and sup of both set A and B are exist.All elements A union B are less than or equal to sup A or sup B.

Then,
exist.
b.We have to prove that

Suppose ,A =(1,2) and B=(2,3)
Sup A=2 , sup B=3

Upper bound of

Hence,

If A=(4,5),B=(2,3)
Sup A=5,Sup B=3


Hence,

Hence, we can say that
.